Question

name
4 - 1 additional practice
solving systems of equations by graphing
use a graph to solve each system of equations. list the solution.

1. {y = 2x - 1, y = -4x - 7}
2. {18x - 3y = 21, y = 6x - 7}
3. {y = 6x + 4, 6x - y = 1}

use a graph to approximate the solution of each system. list the estimated solution.

Explanation:

Step1: Recall the concept of solution of system by graphing

The solution of a system of linear - equations \(y = m_1x + b_1\) and \(y=m_2x + b_2\) is the point \((x,y)\) where the two lines intersect on the graph.

Step2: For the first system \(

$$\begin{cases}y = 2x-1\\y=-4x - 7\end{cases}$$

\)
Set the two equations equal to each other: \(2x-1=-4x - 7\). Add \(4x\) to both sides: \(2x + 4x-1=-7\), which simplifies to \(6x-1=-7\). Then add 1 to both sides: \(6x=-6\), so \(x = - 1\). Substitute \(x=-1\) into \(y = 2x-1\), we get \(y=2\times(-1)-1=-2 - 1=-3\).

Step3: For the second system \(

$$\begin{cases}18x-3y = 21\\y = 6x-7\end{cases}$$

\)
Rewrite the first equation in slope - intercept form \(y=mx + b\). Starting with \(18x-3y = 21\), subtract \(18x\) from both sides: \(-3y=-18x + 21\). Divide by \(-3\): \(y = 6x-7\). Since the two equations are the same, the system has infinitely many solutions.

Step4: For the third system \(

$$\begin{cases}y=6x + 4\\6x-y = 1\end{cases}$$

\)
Rewrite the second equation in slope - intercept form. \(6x-y = 1\) can be rewritten as \(y=6x - 1\). Since the two lines have the same slope (\(m = 6\)) but different \(y\) - intercepts (\(b_1 = 4\) and \(b_2=-1\)), the lines are parallel and there is no solution.

Answer:

1. \((-1,-3)\)
2. Infinitely many solutions
3. No solution